insufficient_budget_no_certify
plain-language theorem explainer
The theorem shows that if per-cycle bandwidth times t is less than 2^n then the accumulated budget falls short of the NP workload for witness size n. Complexity researchers deriving substrate bounds in recognition science cite it to obtain exponential runtime lower bounds on real recognition operators. The proof is a direct term-mode reduction that unfolds the two definitions and applies the hypothesis exactly.
Claim. Let $b=360$ denote the fixed per-cycle BIT bandwidth. If $b t < 2^n$, then the total budget accumulated over $t$ cycles is strictly smaller than the workload $2^n$ required to certify an NP-search witness of size $n$.
background
The module derives structural lower bounds on recognition substrates whose BIT bandwidth is bounded by the recognition operator. Per-cycle bandwidth is fixed at 8 times the consciousness gap, giving the constant 360. Bandwidth budget over t cycles is the product of this per-cycle value and t. NP workload for parameter n is exactly the search-space size 2^n, the minimum number of distinguishable comparisons needed to certify a witness.
proof idea
The proof is a one-line term-mode wrapper. It unfolds bandwidthBudget and npWorkload to expose the product bitBandwidthPerCycle * t and the power 2^n, then invokes the supplied hypothesis directly.
why it matters
This supplies the core implication used by the downstream exponential lower-bound theorem cycles_lower_bound, which rewrites the bandwidth constant and concludes the cycle requirement. It completes the structural lower-bound step in the module derivation: any NP-search problem on a bounded-BIT substrate requires t at least 2^n over B. The result is a phi-rung bound on real recognition systems rather than a classical P-NP separation; the module falsifier is a physical polynomial-time solution on a compatible substrate.
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